3.70.12 \(\int \frac {5+3 x}{90 x+3 x^2+5 x \log (x)} \, dx\)

Optimal. Leaf size=16 \[ 3+\log \left (6+\frac {x}{5}+\frac {\log (x)}{3}\right ) \]

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Rubi [F]  time = 0.19, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {5+3 x}{90 x+3 x^2+5 x \log (x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(5 + 3*x)/(90*x + 3*x^2 + 5*x*Log[x]),x]

[Out]

3*Defer[Int][(90 + 3*x + 5*Log[x])^(-1), x] + 5*Defer[Int][1/(x*(90 + 3*x + 5*Log[x])), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {3}{90+3 x+5 \log (x)}+\frac {5}{x (90+3 x+5 \log (x))}\right ) \, dx\\ &=3 \int \frac {1}{90+3 x+5 \log (x)} \, dx+5 \int \frac {1}{x (90+3 x+5 \log (x))} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.08, size = 10, normalized size = 0.62 \begin {gather*} \log (90+3 x+5 \log (x)) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(5 + 3*x)/(90*x + 3*x^2 + 5*x*Log[x]),x]

[Out]

Log[90 + 3*x + 5*Log[x]]

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fricas [A]  time = 0.80, size = 10, normalized size = 0.62 \begin {gather*} \log \left (3 \, x + 5 \, \log \relax (x) + 90\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x+5)/(5*x*log(x)+3*x^2+90*x),x, algorithm="fricas")

[Out]

log(3*x + 5*log(x) + 90)

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giac [A]  time = 0.17, size = 10, normalized size = 0.62 \begin {gather*} \log \left (3 \, x + 5 \, \log \relax (x) + 90\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x+5)/(5*x*log(x)+3*x^2+90*x),x, algorithm="giac")

[Out]

log(3*x + 5*log(x) + 90)

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maple [A]  time = 0.02, size = 9, normalized size = 0.56

method	result	size
risch	\(\ln \left (\frac {3 x}{5}+\ln \relax (x )+18\right )\)	\(9\)
norman	\(\ln \left (5 \ln \relax (x )+3 x +90\right )\)	\(11\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*x+5)/(5*x*ln(x)+3*x^2+90*x),x,method=_RETURNVERBOSE)

[Out]

ln(3/5*x+ln(x)+18)

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maxima [A]  time = 0.41, size = 8, normalized size = 0.50 \begin {gather*} \log \left (\frac {3}{5} \, x + \log \relax (x) + 18\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x+5)/(5*x*log(x)+3*x^2+90*x),x, algorithm="maxima")

[Out]

log(3/5*x + log(x) + 18)

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mupad [B]  time = 4.26, size = 8, normalized size = 0.50 \begin {gather*} \ln \left (x+\frac {5\,\ln \relax (x)}{3}+30\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*x + 5)/(90*x + 5*x*log(x) + 3*x^2),x)

[Out]

log(x + (5*log(x))/3 + 30)

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sympy [A]  time = 0.13, size = 10, normalized size = 0.62 \begin {gather*} \log {\left (\frac {3 x}{5} + \log {\relax (x )} + 18 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x+5)/(5*x*ln(x)+3*x**2+90*x),x)

[Out]

log(3*x/5 + log(x) + 18)

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